Note on the construction of a pushforward map for differential forms.

1. The Fiber Integral of Differential Forms
Heinrich Hartmann
April 17, 2010
1 Statements
Proposition 1.1. Let f : M → B be an oriented submersion with boundary of
relative dimension d. There is a unique morphism of CN modules
f
: f!Ωd
M/B −→ CN
satisfying
f
α (x) =
f−1(x)
α
for all relative differential forms with proper support α ∈ f!Ωd
M/B.
Moreover there is a natural morphism of CM modules
Φ : Ωk+d
M −→ Ωd
M/N ⊗ f∗
Ωk
N
satisfying
Φ(α ∧ β) → ¯α ⊗ β (1)
for all α ∈ Ωd
M , β ∈ f∗
Ωk
B where ¯α is the image of α in Ωd
M/N . This morphism
induces the a fiber integral
f
: f!Ωk+d
M → Ωk
N .
Definition 1.2. In the situation of the proposition we call f∗ := f
the push
forward or fiber integral of differential forms.
Theorem 1.3. Let f : (M, ∂M) → B be an oriented submersion with boundary
of relative dimension d. The fiber integral has the following properties
1. (Normalisation) If f : M → {pt} then
f∗α =
M
α
1

2. 2. (Fubini-Theorem/Naturality) If g : B → B is another submersion, then
g∗ ◦ f∗ = (f ◦ g)∗
3. (Base Change Formular) If there is a cartesian diagram
M
g
−−−−→ M


f

f
B
g
−−−−→ B.
(2)
where f (and hence f ) is a submersion and g is any differentiable map.
Then
g∗
f∗(α) = f∗g ∗
(α)
for all α ∈ f!(Ωk+d
M ).
4. (Projection Formular) For α ∈ f!(Ωd
M ) and β ∈ Ωk
M )
f∗(α ∧ f∗
β) = (f∗α) ∧ β
5. (Stokes Theorem) Recall that we deonte by ∂f the restriction of f to the
boundray ∂M. For α ∈ f!(Ωk+d
M ) it is
f∗(dα) = (−1)d
d(f∗α) + (f|∂M )∗(α|∂M )
if we use the graded commutator this reads.
[f∗, d] = ∂f
6. If M is a closed manifold and f proper, then f induces a morphism of
cohomology groups
f∗ : Hk
DR(M) → Hk−d
DR (B).
If M and B are compact this map gets identified with the Poincare dual
of the homology pushforward
f∗ : Hm−k(M) → Hm−k(N).
2 Concepts
2.1 Orientation
Definition 2.1. An subersion with boundary f : (M, ∂M) → B is a differen-
tiable map from a manifold with boundary (M, ∂M) to a (ordenary) manifold
B, with such that the restrictions
fo
: Mo
:= M ∂M −→ B, and ∂f : ∂M −→ B
of f to the (ordenary) manifolds Mo
and ∂M are submersions in the usual sense.
2

3. Proposition 2.2 (Existence of fiber space coordinates). Let f : (M, ∂M) → B
be a submersion with boundary of relative dimension d. Then there are open
coverings {Ui}i∈I of M and {Vi}i∈I of B, and open embedings zi : Ui → Rm
+ :=
Rm−1
× R≥0 and yi : Vi → Rb
such that f(Ui) ⊂ Vi and the following diagram
is commutative
Ui
zi
−−−−→ Rm
+

f

p
Vi
bi
−−−−→ Rb
.
(3)
where p is the projectionon the first b coordinates.
Theorem 2.3 (Ehresmann [2] [1]). Let f : (M, ∂M) → B be a proper subersion
with boundary. Then for all b ∈ B there is an open neighbourhood U ⊂ B and
a diffeomorphism
Φ : (M, ∂M)|U −→ (Mb, ∂Mb) × U
with f = Φ ◦ pr2
Definition 2.4. Let ξ : E → M be a vector bundle of rank n. The frame
bundle of E is the principal GLn-bundle
F(ξ) :=
x∈M
Isom(Rn
, Ex)
The orienation bundle Or(E) of E is the fiber bundle associated to F induced
by the GLn-action
GLn −→ Z×
, A → det(A)/| det(A)|
An orientation of E is a global section of Or(E).
Definition 2.5. Let f : M → B be a submersion of relative dimension d. An
orientation of f (or M over B) is an orientation of the relative tangent bundle
TM/B. An orientation of a submersion with boundary f : (M, ∂M) → B is an
orienation of fo
: Mo
→ B.
Remark 2.6. If f : (M, ∂M) → N is an oriented submersion with boundary, then
the fiber f−1
(x) over x ∈ N has a natural structure of an oriented manifold with
boundary.
Recall that an orientation on a manifold with boundary (M, ∂M) is defined
to be an orienation of the interior Mo
and induces a natural orientation of ∂M
(defined using outwards pointing vectorfilds), in the same way an orientation of
f induces an orientation of ∂f.
3 Proofs
Prop. 1.1. The first condition clearly defines f
α as a function on N. It has
just to be checked that this function is smooth. This is a local statement in
3

4. N. Since moreover supp(α) is proper over M we can assume there is a finite
cover of supp(α) admitting fiber space coordinates and a partition of unity
with compact supports on each open subset. The smoothness now follows form
standard results of Analysis in Rn
.
To proof the existence of the morphism Φ we consider the insertion map
ins : Λd
TM/B ⊗ Ωk+d
M −→ Ωk
M .
In the exact sequence (see the appendix)
0 → f∗
Ωk
B → Ωk
M
ϕ
→ f∗
Ω1
B ⊗ Ωk−1
M → Sym2
(f∗
Ω1
B) ⊗ Ωk−2
M → . . .
the arrow ϕ is adjoint to the insertion map f∗
TB ⊗Ωk
M → Ωk−1
M . Hence compo-
sition β ◦ins is adjoint to the insertion d+1 vertical vectorfiedls in a k +d form
and therefore zero. Thus ins factorizes over f∗
Ωk
B. And we get the required
morphism Φ by adjunction.
Note that Φ satisfies the formular 1, since ∧ is itself adjoint to insertion.
The definition of f
is now easy. Application of proper pushforward yields
a map
f!Ωk
M → f!(Ωd
M/B ⊗ f∗
Ωk
B) ∼= f!(Ωd
M/B) ⊗ Ωk
B
which we compose with the above constructed integral. (One needs a proj.
formular for f! ... Reference?/Proof?)
Remark 3.1. In the situation above the morphism Φ can be described very
concretely in coordinates. Let x1
, . . . , xd
, y1
, . . . , yb
be fiber space coordinates
for f : M → B. On the canonical basis Φ takes the values
Φ : dxI
∧ dyJ
→
dxI
⊗ dyJ
if I = [d] := {1, . . . , d}
0 else
It is not hard to check, that this assignment is compatiple with coordinate trans-
formations - and hence can be used as definition. Indeed let ˜x1
, . . . , ˜xd
, ˜y1
, . . . , ˜yb
be another set of fiber space coordinates. We have
dxI
∧ dyJ
=
˜I ˜J
D(˜I, ˜J)d˜x
˜I
∧ d˜y
˜J
where the coefficient D(˜I, ˜J) is the determinant of the matrix
∂xI
/∂˜x
˜I
0
∂xI
/∂˜y
˜J
∂yJ
/∂˜y
˜J
:=













∂xi1
. . . ∂xik
∂yj1
. . . ∂yjl
∂˜xa1
... ∂xip
∂˜xaq 0
∂˜xar
∂˜yb1
... ∂xip
∂˜ybq
∂yjp
∂˜y
˜jq
∂˜ybk













4

5. where of course I = {i1 < · · · < ik}, etc. Note that in general this determinant
is not so easy to compute since the building blocks are not neccessarily diagonal
matrices. However in our case the situation is more fortunate. By definition Φ
takes all forms d˜x
˜I
∧ d˜y
˜J
to zero if not ˜I = [d], but then the first d rows are
lineary dependent if not also I = [n]. So Φ takes the transfomed form to
det(∂xI
/∂˜x
˜I
)d˜x
˜I
⊗
˜J
det(∂yJ
/∂˜y
˜J
)d˜y
˜J
which is precisley the transformation of dxI
⊗ dyJ
as claimed.
Remark 3.2. A submersion determnines a principal fiber bundle on M defined
by
F(f) :=
x∈M
{ϕ : Rd+b ∼=
→ TxM | ϕ(Rd
× {0}n
) ⊂ Txf = ker dfx}
=
x∈M
Isom(Rd
⊂ Rd+b
, Txf ⊂ TxM)
with structure group
GL(d, b) := {invertible block- (d + b) × (d + b) -matrices
A 0
B C
}
= Aut(Rd
⊂ Rd+b
)
All vector bundles occured so far are associated to this principal fiber bundle.
So we could have formulated everything in terms of representation theory for
GL(d, b).
Theorem 1.3. . The Normalisation is obvious form the definition of f∗.
Let us first make the statement of the base change formular precise: Recall
the situation we put ourselfs into:
M
g
−−−−→ M


f

f
B
g
−−−−→ B.
(4)
The fiber integral for f is a morphism of sheaves on B:
f∗ : f!Ωk+d
M −→ Ωk
B.
We can pull back this morphism to B using g and compose with the differential
dg of g, (this is usually deoted by g∗
)
g∗
f!Ωk+d
M
g∗
(f∗)
−→ g∗
Ωk
B
dg
−→ Ωk
B .
5

6. On the other hand g∗
f!Ωk+d
M is canonically isomorphic to f! g ∗
Ωk+d
M . The dif-
ferential dg induces a map
f! g ∗
Ωk+d
M
f!(dg )
−→ f! Ωk+d
M
which we can compose with the fiber integral f∗. The Base Change formular
claims now, that this two morphisms concide, i.e. the following diagram of
sheaves on B commutes:
f! g∗Ωk+d
M
g∗
(f∗)
−−−−→ g∗
Ωk
B
f! dg

 g∗


f! Ωk+d
M
(f∗)
−−−−→ Ωk
B .
By definition of the fiber integral the rows are the proper direct image f! of
morphism on M which we can further decompose... use adjunction properties
of f!
, f∗, f∗
in the category of ringed spaces...
We can also prove this directly in coordinates: Let b ∈ B set b := g(b )0.
take α ∈ (g∗
f!Ωk+d
M )b, that is a form defined on an open subset f−1
(U) ⊂ M
where U is an arbitary small neighbourhood of b in B, which has proper support
over B.
By Ehresmanns theorem can assume there is a fiber preserving diffeomor-
phism
MU
Ψ
−→ Mb × U
and coodrdinates yj
: U → R. We now choose a cover of Mb admitting co-
ordinates xi
on each patch V ⊂ Mb and a partition of unity with compact
supports.
By definition of fiber square M is canonically isomorphic to B ×g M, hence
f−1
(g−1
(U)) is canonically isomorphic to g−1
(U)×Mb. After making U smaller,
we can again assume there are coordinates y j
on g−1
(U).
So we have reduced the statement to the affine situation1
. Let
α =
I,J
αIJ (x, y)dxI
∧ dyJ
be a k + d-form on Rd
+ × Rb
proper over Rb
.
g : Rb
→ Rb
be a differentiable map. Then
|J|=k
( α[d]J (x, y)dx1
∧ · · · ∧ dxd
) ∧ dyJ
...
1A little attention has to be paid to the orientation. Either we assume that the chart
transions of Mb are orientation preserving, or we multiply by dxI by Or(∂/∂x1, . . . , ∂/∂xd)
before going to the affine case.
6

7. 4 Appendix
4.1 Exterior and Symmetric Algebras
We recall some baisc facts about exterior algebras.
Let V be a finite dimensional vector space over a field K. Set
T(V ) =
k≥0
V ⊗k
be the Tensor algebra over V . As T is functorial in V we get a canonical action
GL(V ) on T(V ) by Algebra Homomorphisms. The action of the subgroup
K×
⊂ GL(V ) induces a N0-grading on T(V ) which coincedes with the grading
we used in the definition (by k ≥ 0).
The exterior and symmetric algebras can be constructed out of T(V ) in two
ways:
4.1.1 Construction as a quotient
Let I be the two sided ideal of T(V ) generated by the elements
{x ⊗ y + y ⊗ x | x, y ∈ V }
Then Λ(V ) := T(V )/I is the exterior algebra. As I is a graded ideal the grading
of T(V ) induces a natural grading on Λ∗
(V ). It is I moreover invariant for the
action of GL(V ), so Λ(V ) carries a natural GL(V ) action, too.
We denote the image of x1 ⊗ . . . ⊗ xk in Λ(V ) by x1 ∧ · · · ∧ xk.
For the symmetric algebra Symk
(V ) we exchange x⊗y+y⊗x with x⊗y−y⊗x
above. The same statements hold mutatis mutandis in this case.
We denote the image of x1 ⊗ . . . ⊗ xk in Sym(V ) by x1 · · · xk.
We now turn to the second description
4.1.2 Construction as a subspace
There is a natural Sk = AutSET({1, . . . , k}) left action on Tk
(V ) given by
τ : x1 ⊗ . . . ⊗ xk → xτ1 ⊗ . . . ⊗ xτk
for τ ∈ Sk.
Note that if we embed Sk × Sl into Sk+l by letting Sk act on {1, . . . , k} and
Sl on {k + 1, . . . , l} in the obvious way, then the product
Tk
(V ) ⊗ Tl
(V ) → Tk+l
(V )
is Sk × Sl-equivariant.
For a character χ : Sk → Z×
define an operator Pχ : T(V ) → T(V ) by
x1 ⊗ . . . ⊗ xk →
1
k!
τ∈Sk
χ(τ) τ · (x1 ⊗ . . . ⊗ xk).
7

8. One checks easily that Pχ is a projection operator i.e. P2
χ = Pχ. Hence we have
a canonical decomposition:
Tk
(V ) = Image(Pχ) ⊕ Kernel(Pχ)
which is invariant for the Sk-action as well as for the GL(V ) action on Tk
(V )!
We say S = P1 is the symmetrisation operator and A = Psign is the anti-
symmetrisation operator. In degree k ≥ 2 we have S ◦ A = A ◦ S = 0. We
denote the image of A and S by Ak
(V ) resp. Sk
(V ).
Proposition 4.1. The compositions
A(V ) → T(V ) → Λ(V )
and
S(V ) → T(V ) → Sym(V )
are natural isomorphisms of vectorspaces, GL(V )-representations and induce
isomorphisms of Sk-representations in the k-th graded component.
These isomorphisms can be made explicit as follows: Take x1 ∧ · · · ∧ xk ∈
Λ(V ) represent it by some (non-anti-symmetric) tensor x1 ⊗ . . . ⊗ xk. Then the
anti-symmetrisation A(x1 ⊗ . . . ⊗ xk) lies in A(V ) and is an inverse image for
x1 ∧ · · · ∧ xk.
4.1.3 An exact sequence
Let
0 −→ S −→ V −→ Q −→ 0 (5)
be an exact sequence of finite dimensional k-vector spaces. The induced seqences
0 −→ Symk
(S) −→ Symk
(V ) −→ Symk
(Q) −→ 0
and
0 −→ Λk
(S) −→ Λk
(V ) −→ Λk
(Q) −→ 0
are exact at the right and left, but not in the middle. Instead we have natural
(longer) exact sequences
0 → Symk
(S) → Symk
(V ) → Symk−1
(V ) ⊗ Q → . . . (6)
. . . → Sym2
(V ) ⊗ Λk−2
(Q) → V ⊗ Λk−1
(Q) → Λk
(Q) → 0
0 → Λk
(S) → Λk
(V ) → Λk−1
(V ) ⊗ Q → . . . (7)
. . . → Λ2
(V ) ⊗ Symk−2
(Q) → V ⊗ Symk−1
(Q) → Symk
(Q) → 0
and the dual versions
0 → Symk
(S) → V ⊗ Symk−1
(S) → Λ2
(V ) ⊗ Symk−2
(S) → . . .
. . . → Λk−1
(V ) ⊗ S → Λk
(V ) → Λk
(Q) → 0
8

9. 0 → Λk
(S) → V ⊗ Λk−1
(S) → Sym2
(V ) ⊗ Λk−2
(S) → . . .
. . . → Symk−1
(V ) ⊗ S → Symk
(V ) → Symk
(Q) → 0.
First of all we have to construct natural morphisms
Symk
(V ) ⊗ Λl
(Q) → Symk−1
(V ) ⊗ Λl+1
(Q)
Λk
(V ) ⊗ Syml
(Q) → Λk−1
(V ) ⊗ Syml+1
(Q)
and
Symk
(S) ⊗ Λl
(V ) → Symk−1
(S) ⊗ Λl+1
(V )
Λk
(S) ⊗ Syml
(V ) → Λk−1
(S) ⊗ Syml+1
(V )
To cunstruct the first morphism we compose the tensored product of the natural
embedings
Symk
(V ) → Tk
(V ), Λl
(Q) → Tk
(Q)
with the canonical map
Tk
(V ) ⊗ Tl
(Q) → Tk−1
(V ) ⊗ Tl+1
(Q)
mapping the last V variable to its image in Q. Now we use the quotient desc-
tription of Λ and Sym to end up with the required morphism. The remaining
morphism are defined similary.
The vanishing of the composition
Symk+1
(V ) ⊗ Λl−1
(Q) → Symk
(V ) ⊗ Λl
(Q) → Symk−1
(V ) ⊗ Λl+1
(Q)
and the exactnes can be checked in coordinates - it holds true basically because
S ◦ A = A ◦ S = 0 and the ideals are generated in degree 2.
4.2 Multiplication Formular
Let f : M → N be a differentiable map of smooth manifolds with local
coordinates xi
, i = 1 . . . m and yi
, i = 1, . . . , n respectively. For two Index
sets I ⊂ [m] := {1, . . . , m}, J ⊂ [n] of cardinality k. we define the matrix
∂yJ
∂xI := (∂yj
◦f
∂xi )j∈J,i∈I where we order the indices in ascending order.
Lemma 4.2. With this notation we have:
dyJ
= dyj1
∧ · · · ∧ dyjk
=
I⊂[m],|I|=k
det(
∂yJ
∂xI
)dxI
where of course j1 < · · · < jk.
Proof. We have
dyJ
=
i1=1..k
· · ·
ik=1..k
∂yj1
∂xi1
dxi1
∧ · · · ∧
∂yjk
∂xik
dxik
9

10. if in this sum ir = is for some r = s then the corresponding summand is zero.
Hence the set {i1, . . . , ik} ⊂ [m] can be assumed to have cardinality k. There is
now an unique permutation π ∈ Sk such that iπ1 < · · · < iπk. We further note
that
dxi1
∧ · · · ∧ dxik
= sign(π)dxiπ1
∧ · · · ∧ dxiπk
.
Combining this with the previews formular yields:
dyJ
=
I⊂[m],|I|=k π∈Sk
sign(π)
∂yj1
∂xiπ1
. . .
∂yjk
∂xiπk
dxI
=
I⊂[m],|I|=k
det(
∂yJ
∂xI
)dxI
where now i = {i1, . . . , ik} with i1 < · · · < ik.
4.3 Localised Manifolds (obsolete)
Definition 4.3. A localised manifold M = (CM , AM ) is a manifold CM together
with a subset AM ⊂ CM . We call CM the craig and AM the core of M.
A morphism of localised manifolds f : (CM , AM ) → (CN , AN ) is an equica-
lence class of diffeneriable mappings
Cf : D(Cf ) → N with f(AM ) ⊂ AN
where D(Cf ) is an open neighbourhood of AM in CM . Two maps Cf and Cg are
equivalent iff there is an open neighbourhood V of A contained in D(Cf )∩D(Cg)
and
f|V = g|V .
If f : M → M and g : M → M are morphisms of localised manifolds, we
define the composition by composing representatives 2
. So we get a category of
localised Manifolds which we will denote by LDIFF.
The dimension M is the dimension of CM , which is preserved under isomor-
phisms.
Question: Is the functor LDIFF → LkRSp which maps (M, A) to i−1
(CM )
where i : M → A fully faithful?
What about HomLDIFF(Rn
, 0) → (Rn
, 0))? It works?!
If not, is it the restriction to mfds. with boundary f.f. ?
Yes, then we can skip all this, and define mf. with boundary as lk-ringed
spaces.
Remark 4.4. If AM ⊂ CM is open, then (CM , AM ) ∼= (AM , AM ) as localised
manifolds.
Definition 4.5. Let M = (CM , AM ) be a localised manifold, an open sub-
manifold of M is the localised manifold (U, AM ∩ U) given by an open subset
2The set of definition D(Cf ) can be made smaller to ensure the latter morphism is defined
on the image
10

11. U ⊂ CM of the craig. We have an obvious morphism of localised manifolds
(U, AM ∩ U) → M.
An open cover of M is a collection {Ui}i∈I of open submanifolds such that
AUi
cover AM .
Remark 4.6. We have a localisation functor
L : DIFF → LDIFF, M → (M, M), f → f
which is full and faithful. And a restriction functor
A : LDIFF → TOP, (CM , AM ) → AM , f → Af = f|AM
: AM → AN .
which is neither faithful nor full.
Definition 4.7. A morphism f : M → N of localied manifolds is called submer-
sion/immersion if there exists an representative Cf which has the correspondig
property.
f is called injective/surjective/proper/open/closed, if Af : AM → AN has
the correspondig property.
We call f an open embeding, if f is open and induces an isomorphism of
localised manifolds M → (N, Image(Af )).
Definition 4.8. A manifold with bounday is a localised manifold M locally
isomorphic to Rn
+ := (Rn
, Rn−1
× R≥0), i.e. there is an open cover {Ui}i∈I of
M and open embedings ϕi : Ui → Rn
+ of localised manifolds.
Let (CM , AM ) be a manifold with boundary. We denote the topological
boundary of AM ⊂ CM by ∂M. As AM is closed in CM we have ∂M ⊂ AM
and ∂M is hence preserved under isomorphisms of localised manifolds. The
boundary ∂M carries a natural structure of an (ordenary) submanifold of CM .
Moreover Mo
:= AM ∂M is a (orendenary) manifold, too.
We also use the notation (M, ∂M) for a manifold with boundary, which
means M = (CM , AM ) is a manifold with boundary and ∂M ⊂ AM its bound-
ary.
Definition 4.9. A submersion of manifolds with boundary is a submersion of
a localised manifolds f : M → B, where B = (B, B) is an (ordenary) manifold
and M = (CM , AM ) a manifold with bounday, such that ∂f, the restriction of
f to ∂M, is still submersive.
The condition that ∂f is submersive is not automatic. Let for example be
D ⊂ R2
be the closed disc, then
pr1 : (R2
, D) → (R, R)
is clearly a submersion of localised manifolds but not submersive on ∂D.
Remark 4.10. In the situation of the definition, let U ⊂ M be an open subset.
Then the restriction of π to U, M ∩ U is again a submersion with boundary.
11

12. Proposition 4.11. Let M → B be a submersion of manifolds with boundary,
dim(M) = m, dim(B) = b. Then locally on M there are fiber space coordinates.
This means there are covers {Ui}i∈I of M and {Vi}i∈I of B, and open embedings
zi : Ui → Rm
+ and bi : Vi → Rb
such that f(Ui) ⊂ Vi and the following diagram
is commutative
Ui
zi
−−−−→ Rm
+

f

p
Vi
bi
−−−−→ Rb
.
(8)
Here p is the projection to the first b corrdinates.
Definition 4.12. A sheaf on a localised manifold M is a sheaf on AM . (Is this
equivalent to sheafs on the Site of open covers?)
The structure sheaf of M is i−1
(CCM
)
The tangent sheaf of M is i−1
(TCM
) where i : AM → CM is the inclusion
and TCM
the Tangent sheaf of CM .
The relative tangent sheaf of a morphism f : M → N of localised manifolds
is...
References
[1] C. Ehresmann. Sur les espaces fibres differentiables. CR Acad. Sci. Paris,
224:1611–1612, 1947.
[2] J.A. Wolf. Differentiable fibre spaces and mappings compatible with Rie-
mannian metrics. Michigan Math. J, 11(1):65–70, 1964.
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